Invariance refers to the preservation of something: intervals, dynamics, rhythms, pitches, and so on. In elementary twelve-tone theory, we are mostly concerned with intervallic invariance and pitch class segmental invariance.
Intervallic Invariance
Any time a row is transposed, the ordered intervallic content of the row is unchanged. Thus, transposition always results in intervallic invariance. Retrograde inversion creates retrograde intervallic invariance.
Segmental Invariance
When a pitch-class segment of a row is unchanged when that row is transformed, we say that the segment is “held invariant.” Consider the following example, from Webern’s String Quartet, Op. 28:
The brackets show the discrete tetrachords of the row. Notice that these tetrachords are the same amongst the to different rows. That is, the tetrachords are invariant segments. These segments are held invariant because of they share the same relationship with one another that the rows share. Because the tetrachords are related by T8, when the row as a whole is transposed by T8, those tetrachords are “held invariant.” (Think of the process like this: when the first tetrachord [6789] is transposed by T8, it becomes the last tetrachord [2345]. And therefore, when the whole row is transposed by T8, _the last tetrachord _becomes the first tetrachord.)
To determine when and if a pitch-class segment of a row will be held invariant:
(1) Find an equivalent set-class elsewhere in the row. This may be a dyad, trichord, tetrachord, etc.
(2) Determine the transpositional or inversional relationship between them.
(3) When the row is transposed or inverted by that same relationship a segment will be held invariant.